I'm not a certified welder, but I did take a few manufacturing labs as an engineering undergrad. Look into welding certification and training in your area, and you might find a tech school that offers some good instruction. You can probably get by with a good reference book and a lot of practice, though. You can try starting out on Stick welding (formally known as SMAW). This type of welder is usually the most inexpensive, and most models let you reverse the polarity easily if you want to run vertical or overhead welds (not that you would start with those, but it might be useful eventually.) You need to pause often to change electrodes with SMAW. The weld puddle is shielded from the atmosphere by means of the flux on the electrode.
Much of what I've seen welded is done on MIG welders (formally known as GMAW), which are fairly easy to operate. They use a metal inert gas (hence the acronym MIG) to shield the weld puddle as it solidifies. You will need to keep a tank of inert gas handy, usually Argon or CO2, or your welds will be very poor. Some MIG welders will accept flux-core wire, which works very similar to stick welding, but you don't have to pause and change electrodes and you don't use a shielding gas (very convenient!). It spits a bit more, and you can't weld small parts with it, but it is what I keep in my basement shop for repairs. This is what I would recommend for you as well, because it is very versatile and reasonably affordable.
TIG (GTAW) can be used for thin materials, since you have much more control over the current. It takes a lot of practice compared to the other kinds of welding, but produces beautiful welds on materials that otherwise cannot be welded conventionally.
Whatever you choose, good luck! I would recommend staying away from acetylene at first, because it's incredibly dangerous. I use flux-core, but it's a matter of personal choice. As long as you prepare the metal properly, you shouldn't have any trouble. Hope this helps!